The question I'm considering is the following: given an 1-cocycle in of the modular group in Hom$(H;\textrm{GL}_{r}(C))$ call it $f$ when does it induce a vector bundle structure on the corresponding curve $X(1)$ (or more genrally $X(\Gamma)$)?

The reason why I'm confused over this is because I have two contradictory views on why it should hold in general/not: This first is baased around a similar intuition to filling in punctured Riemann surface - namely we can clearly define a vector bundle over $X(\Gamma)$ without the cusps & points of non-trivial stabilizers. As these points form a finite set we can choose suitable charts for them such that the cocycle doesn't need to be defined at these points so we can just take it to be as for the simpler quotient space.

The competing reasoning is by example - if we look at a non-trivial 1-dim rep of the modular group say $S \mapsto i$ and $T \mapsto -\overline{\rho}$ then this gives us a 1-cocycle (that is independant of $\tau$) and thus would induce a line bundle on $X(1)$. But as this line bundle would have it's 6th power isomorphic to the canonical line bundle this isn't possible.

I don't really feel which of these statements is wrong - any ideas would be musch appreciated.